(a) State Ampere's circuital law. Use this law to obtain the expression for the magnetic field inside an air-cored toroid of average radius r, having n turns per unit length and carrying a steady current I. (b) An observer to the left of a solenoid of N turns, each of cross-sectional area A, observes that a steady current I flows in the clockwise direction. Depict the magnetic field lines due to the solenoid, specify its polarity, and show that it acts as a bar magnet of magnetic moment m=NIA.
Answer: (a) B=μ0nI. (b) The observed face is a south pole; m=NIA.
Solution
(a) \textbf{Ampere's law:} ∮B⋅dℓ=μ0Ienc. For a circular loop of radius r inside the toroid, B(2πr)=μ0(2πrn)I, so
B=μ0nI.
(b) A clockwise current as seen by the observer makes the near face a \textbf{south pole} (field lines enter there) and the far face a north pole, so the field lines run through the core like those of a bar magnet. Each turn is a dipole of moment IA; the N turns in series give a total moment m=NIA along the axis, so the solenoid behaves as a bar magnet of moment NIA.
